Gleason-Kahane-\.Zelazko theorems in function spaces

TL;DR

This paper surveys recent extensions of the Gleason-Kahane-Żelazko theorem to Banach function spaces that are not algebras, highlighting new developments in functional analysis.

Contribution

It provides a concise overview of recent generalizations of the theorem beyond Banach algebras to broader function spaces.

Findings

Extensions of the theorem to non-algebra Banach function spaces

Characterization of linear functionals non-zero on invertible elements

Summary of recent research developments

Abstract

The Gleason-Kahane-\.Zelazko theorem states that a linear functional on a Banach algebra that is non-zero on invertible elements is necessarily a scalar multiple of a character. Recently this theorem has been extended to certain Banach function spaces that are not algebras. In this article we present a brief survey of these extensions.